Next upgrade to an entry-level system?

I would add another subwoofer and then as mentioned the phono stage. 

I agree with other posters about upgrading the speakers next. There are quite a few well regarded speakers under 1K new. I followed a similar upgrade path and the new speakers helped reveal limitations with my other components. 
The next step would be the amp. Rotel, Cambridge Audio and Rega have well reviewed integrated amps under 1K. There’s real savings in the used market if you feel comfortable buying 2nd hand.

Speakers, and quality stands.  From your question and screen name, I take it you are a classical music lover. Klipsch speakers sound signature is not what I prefer for classical music. If I were in your position I would search out FBM, USAM, and here for a pair of used ADS L1290/2, L730, L780/2, or L880/2.  Make sure they have original drivers and .75” tweeters. All these models have ample midbass power and smooth extended highs.  Great clarity with no glare.  These are not dull New England AR type speakers…they are European inspired and were built to last.  The crossovers used mostly film caps and air core inductors. Rubber surrounds. Soft fabric domes with “sticky” damping compound. 
If any restoration is needed, Richard So does it best. 

AT about $1K, I would give a hard look at the Paradigm Premiere 200b, which often go on sale at around a bit over $800.  I have a pair driven by a Rega Brio in my second (bedroom) system, and they sound wonderful; my soul source for this system is streaming.  Excellent imaging for this price range, with a good balance across the frequency spectrum, although your sub will add some needed punch in the lower end.  I am quite pleased with them.  Worth a look and listen if you can, and they are manufactured in Canada.